How does sound dampening foam work?

The sound dampening is an effect of the foam being placed on the case panels, not against the fans. How much good it does can vary from moderate to barely noticeable. It also depends on the type of foam. Some use a closed cell, some use an open cell foam (think soft packing foam like an electronics case, not foam peanuts) - others use a dense foam more like rubber similar to dynomat (a heavy rubber material applied usually to the inside of autos to help keep body panels from resonating from high levels of noise/bass).

Some foams have both a thicker membrane against the case panel topped with a soft open cell foam. It's usually a rather thin layer, around 1/4-3/8" thick. It can work in a couple of ways. Sound resonates (whether hdd seeking sounds/vibrations, air noise from fans etc) against thin metal panels and this can increase the noise factor. By having the rubbery membrane against the case panels, it 'deadens' the resonating qualities of the metal panels. Rather than 'ringing' the sound off them, it will create more of a 'thud'. Imagine letting a tuning fork ring freely vs silencing it with your fingers.

The other way the foam works is for surrounding noise such as from fans, when sound hits it the foam has a tendency to absorb the sound due to the imperfect surface of the foam. Lots of little waves and valleys in the foam trap sound waves and interrupt them. You always want gaps around everything to ensure airflow. The usual places foam is placed is against the inside of the side panels, top panel of the case (barring any airflow obstruction), the bottom panel of the case, the inside of the front panel if you can access it (like just inside a front panel door). Think of the wavy acoustic foam used in sound studios which when position properly can nearly sound proof a room. (often used in music practice rooms in universities and performs well enough that someone can be playing an upright piano in the next room and you'd never hear it through an adjoining wall).

Keep in mind a pc isn't a sealed box, any opening will allow sound to escape (such as intake and exhaust fans). Fans themselves have more of an impact on this, quieter, better quality fans will quiet down a pc far better than foam will. In order to quiet fans, many people use flexible rubber 'pins' that snap against the fan connection holes rather than screws which hold a fan tightly to a point on the case - sort of decoupling them so any vibration is lost in the rubber fan mounts rather than transferred through the case or components.